\(\int (d+e x)^2 \log (c (a+b x^3)^p) \, dx\) [192]

Optimal result

Integrand size = 20, antiderivative size = 250 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3-\frac {\sqrt {3} \sqrt [3]{a} d \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 b^{2/3}}-\frac {\left (b d^3-a e^3\right ) p \log \left (a+b x^3\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e} \]

[Out]

-3*d^2*p*x-3/2*d*e*p*x^2-1/3*e^2*p*x^3+a^(1/3)*d*(b^(1/3)*d-a^(1/3)*e)*p*ln(a^(1/3)+b^(1/3)*x)/b^(2/3)-1/2*a^(
1/3)*d*(b^(1/3)*d-a^(1/3)*e)*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(2/3)-1/3*(-a*e^3+b*d^3)*p*ln(b*x^3
+a)/b/e+1/3*(e*x+d)^3*ln(c*(b*x^3+a)^p)/e-a^(1/3)*d*(b^(1/3)*d+a^(1/3)*e)*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a
^(1/3)*3^(1/2))*3^(1/2)/b^(2/3)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2513, 1850, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {\sqrt [3]{a} d p \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 b^{2/3}}-\frac {\sqrt {3} \sqrt [3]{a} d p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right )}{b^{2/3}}+\frac {\sqrt [3]{a} d p \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}-\frac {p \left (b d^3-a e^3\right ) \log \left (a+b x^3\right )}{3 b e}-3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3 \]

[In]

Int[(d + e*x)^2*Log[c*(a + b*x^3)^p],x]

[Out]

-3*d^2*p*x - (3*d*e*p*x^2)/2 - (e^2*p*x^3)/3 - (Sqrt[3]*a^(1/3)*d*(b^(1/3)*d + a^(1/3)*e)*p*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(2/3) + (a^(1/3)*d*(b^(1/3)*d - a^(1/3)*e)*p*Log[a^(1/3) + b^(1/3)*x])/b^(2
/3) - (a^(1/3)*d*(b^(1/3)*d - a^(1/3)*e)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(2/3)) - ((b*d
^3 - a*e^3)*p*Log[a + b*x^3])/(3*b*e) + ((d + e*x)^3*Log[c*(a + b*x^3)^p])/(3*e)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1850

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)
/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1901

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 2513

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[(f
+ g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n)^p])/(g*(r + 1))), x] - Dist[b*e*n*(p/(g*(r + 1))), Int[x^(n - 1)*((f
+ g*x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}-\frac {(b p) \int \frac {x^2 (d+e x)^3}{a+b x^3} \, dx}{e} \\ & = -\frac {1}{3} e^2 p x^3+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}-\frac {p \int \frac {x^2 \left (3 \left (b d^3-a e^3\right )+9 b d^2 e x+9 b d e^2 x^2\right )}{a+b x^3} \, dx}{3 e} \\ & = -\frac {1}{3} e^2 p x^3+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}-\frac {p \int \left (9 d^2 e+9 d e^2 x-\frac {3 \left (3 a d^2 e+3 a d e^2 x-\left (b d^3-a e^3\right ) x^2\right )}{a+b x^3}\right ) \, dx}{3 e} \\ & = -3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}+\frac {p \int \frac {3 a d^2 e+3 a d e^2 x-\left (b d^3-a e^3\right ) x^2}{a+b x^3} \, dx}{e} \\ & = -3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}+\frac {p \int \frac {3 a d^2 e+3 a d e^2 x}{a+b x^3} \, dx}{e}-\frac {\left (\left (b d^3-a e^3\right ) p\right ) \int \frac {x^2}{a+b x^3} \, dx}{e} \\ & = -3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3-\frac {\left (b d^3-a e^3\right ) p \log \left (a+b x^3\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}+\frac {p \int \frac {\sqrt [3]{a} \left (6 a \sqrt [3]{b} d^2 e+3 a^{4/3} d e^2\right )+\sqrt [3]{b} \left (-3 a \sqrt [3]{b} d^2 e+3 a^{4/3} d e^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} \sqrt [3]{b} e}-\frac {\left (\left (-3 a \sqrt [3]{b} d^2 e+3 a^{4/3} d e^2\right ) p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} \sqrt [3]{b} e} \\ & = -3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\left (b d^3-a e^3\right ) p \log \left (a+b x^3\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}-\frac {\left (\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{2/3}}+\frac {1}{2} \left (3 a^{2/3} d \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx \\ & = -3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 b^{2/3}}-\frac {\left (b d^3-a e^3\right ) p \log \left (a+b x^3\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e}+\frac {\left (3 \sqrt [3]{a} d \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{2/3}} \\ & = -3 d^2 p x-\frac {3}{2} d e p x^2-\frac {1}{3} e^2 p x^3-\frac {\sqrt {3} \sqrt [3]{a} d \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 b^{2/3}}-\frac {\left (b d^3-a e^3\right ) p \log \left (a+b x^3\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.19 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.92 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {-\frac {p \left (18 b d^2 e x+9 b d e^2 x^2+2 b e^3 x^3+6 \sqrt {3} \sqrt [3]{a} b^{2/3} d^2 e \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-9 b d e^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )-6 \sqrt [3]{a} b^{2/3} d^2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+3 \sqrt [3]{a} b^{2/3} d^2 e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \left (b d^3-a e^3\right ) \log \left (a+b x^3\right )\right )}{2 b}+(d+e x)^3 \log \left (c \left (a+b x^3\right )^p\right )}{3 e} \]

[In]

Integrate[(d + e*x)^2*Log[c*(a + b*x^3)^p],x]

[Out]

(-1/2*(p*(18*b*d^2*e*x + 9*b*d*e^2*x^2 + 2*b*e^3*x^3 + 6*Sqrt[3]*a^(1/3)*b^(2/3)*d^2*e*ArcTan[(1 - (2*b^(1/3)*
x)/a^(1/3))/Sqrt[3]] - 9*b*d*e^2*x^2*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)] - 6*a^(1/3)*b^(2/3)*d^2*e*Lo
g[a^(1/3) + b^(1/3)*x] + 3*a^(1/3)*b^(2/3)*d^2*e*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*(b*d^3 - a
*e^3)*Log[a + b*x^3]))/b + (d + e*x)^3*Log[c*(a + b*x^3)^p])/(3*e)

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.34

method	result	size
parts	\(\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e^{2} x^{3}}{3}+\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e d \,x^{2}+d^{2} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) x +\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) d^{3}}{3 e}-\frac {p b \left (\frac {e \left (\frac {1}{3} e^{2} x^{3}+\frac {3}{2} d e \,x^{2}+3 d^{2} x \right )}{b}+\frac {-3 e \,d^{2} a \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 e^{2} d a \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a \,e^{3}+b \,d^{3}\right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b}\right )}{e}\)	\(336\)
risch	\(\frac {\left (e x +d \right )^{3} \ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{3 e}-\frac {i x \pi \,d^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}+\frac {i x \pi \,d^{2} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}-\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}+\frac {i x \pi \,d^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}+\frac {i e^{2} \pi \,x^{3} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{6}+\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}+\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{6}+\frac {i e \pi d \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}-\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{6}-\frac {i x \pi \,d^{2} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}-\frac {i e^{2} \pi \,x^{3} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{6}-\frac {i e \pi d \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {\ln \left (c \right ) e^{2} x^{3}}{3}-\frac {e^{2} p \,x^{3}}{3}+e \ln \left (c \right ) d \,x^{2}-\frac {3 d e p \,x^{2}}{2}+\ln \left (c \right ) d^{2} x -3 d^{2} p x +\frac {p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\left (a \,e^{3}-b \,d^{3}\right ) \textit {\_R}^{2}+3 e^{2} d a \textit {\_R} +3 e \,d^{2} a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b e}\)	\(537\)

[In]

int((e*x+d)^2*ln(c*(b*x^3+a)^p),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(c*(b*x^3+a)^p)*e^2*x^3+ln(c*(b*x^3+a)^p)*e*d*x^2+d^2*ln(c*(b*x^3+a)^p)*x+1/3*ln(c*(b*x^3+a)^p)/e*d^3-p*
b/e*(e/b*(1/3*e^2*x^3+3/2*d*e*x^2+3*d^2*x)+(-3*e*d^2*a*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*
ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-3*e^2*d*a
*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)
^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))+1/3*(-a*e^3+b*d^3)/b*ln(b*x^3+a))/b)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.32 (sec) , antiderivative size = 5799, normalized size of antiderivative = 23.20 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^2*log(c*(b*x^3+a)^p),x, algorithm="fricas")

[Out]

Too large to include

Sympy [A] (verification not implemented)

Time = 15.87 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.69 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=3 a d^{2} p \operatorname {RootSum} {\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} + 3 a d e p \operatorname {RootSum} {\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )} + \frac {a e^{2} p \left (\begin {cases} \frac {x^{3}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x^{3} \right )}}{b} & \text {otherwise} \end {cases}\right )}{3} - 3 d^{2} p x + d^{2} x \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {3 d e p x^{2}}{2} + d e x^{2} \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {e^{2} p x^{3}}{3} + \frac {e^{2} x^{3} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{3} \]

[In]

integrate((e*x+d)**2*ln(c*(b*x**3+a)**p),x)

[Out]

3*a*d**2*p*RootSum(27*_t**3*a**2*b - 1, Lambda(_t, _t*log(3*_t*a + x))) + 3*a*d*e*p*RootSum(27*_t**3*a*b**2 +
1, Lambda(_t, _t*log(9*_t**2*a*b + x))) + a*e**2*p*Piecewise((x**3/a, Eq(b, 0)), (log(a + b*x**3)/b, True))/3
- 3*d**2*p*x + d**2*x*log(c*(a + b*x**3)**p) - 3*d*e*p*x**2/2 + d*e*x**2*log(c*(a + b*x**3)**p) - e**2*p*x**3/
3 + e**2*x**3*log(c*(a + b*x**3)**p)/3

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{6} \, b p {\left (\frac {2 \, e^{2} x^{3} + 9 \, d e x^{2} + 18 \, d^{2} x}{b} - \frac {6 \, \sqrt {3} {\left (a b d e \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{2}} - \frac {{\left (2 \, a e^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a d e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a d^{2}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (a e^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a d e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a d^{2}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \]

[In]

integrate((e*x+d)^2*log(c*(b*x^3+a)^p),x, algorithm="maxima")

[Out]

-1/6*b*p*((2*e^2*x^3 + 9*d*e*x^2 + 18*d^2*x)/b - 6*sqrt(3)*(a*b*d*e*(a/b)^(2/3) + a*b*d^2*(a/b)^(1/3))*arctan(
1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2) - (2*a*e^2*(a/b)^(2/3) + 3*a*d*e*(a/b)^(1/3) - 3*a*d^2)*l
og(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) - 2*(a*e^2*(a/b)^(2/3) - 3*a*d*e*(a/b)^(1/3) + 3*a*d^2
)*log(x + (a/b)^(1/3))/(b^2*(a/b)^(2/3))) + 1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)*log((b*x^3 + a)^p*c)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.10 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{3} \, {\left (e^{2} p - e^{2} \log \left (c\right )\right )} x^{3} + \frac {a e^{2} p \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac {1}{2} \, {\left (3 \, d e p - 2 \, d e \log \left (c\right )\right )} x^{2} - {\left (3 \, d^{2} p - d^{2} \log \left (c\right )\right )} x + \frac {1}{3} \, {\left (e^{2} p x^{3} + 3 \, d e p x^{2} + 3 \, d^{2} p x\right )} \log \left (b x^{3} + a\right ) + \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b d^{2} p - \left (-a b^{2}\right )^{\frac {2}{3}} d e p\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2}} - \frac {{\left (a b d e p \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b d^{2} p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b d^{2} p + \left (-a b^{2}\right )^{\frac {2}{3}} d e p\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{2}} \]

[In]

integrate((e*x+d)^2*log(c*(b*x^3+a)^p),x, algorithm="giac")

[Out]

-1/3*(e^2*p - e^2*log(c))*x^3 + 1/3*a*e^2*p*log(abs(b*x^3 + a))/b - 1/2*(3*d*e*p - 2*d*e*log(c))*x^2 - (3*d^2*
p - d^2*log(c))*x + 1/3*(e^2*p*x^3 + 3*d*e*p*x^2 + 3*d^2*p*x)*log(b*x^3 + a) + sqrt(3)*((-a*b^2)^(1/3)*b*d^2*p
- (-a*b^2)^(2/3)*d*e*p)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^2 - (a*b*d*e*p*(-a/b)^(1/3) +
a*b*d^2*p)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b) + 1/2*((-a*b^2)^(1/3)*b*d^2*p + (-a*b^2)^(2/3)*d*e*p
)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^2

Mupad [B] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.43 \[ \int (d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,b^3\,c^3-27\,a\,b^2\,c^2\,e^2\,p+81\,a\,b^2\,c\,d^3\,e\,p^2+9\,a^2\,b\,c\,e^4\,p^2-27\,a\,b^2\,d^6\,p^3-a^3\,e^6\,p^3,c,k\right )\,\left (\mathrm {root}\left (27\,b^3\,c^3-27\,a\,b^2\,c^2\,e^2\,p+81\,a\,b^2\,c\,d^3\,e\,p^2+9\,a^2\,b\,c\,e^4\,p^2-27\,a\,b^2\,d^6\,p^3-a^3\,e^6\,p^3,c,k\right )\,a\,b^2\,9-6\,a^2\,b\,e^2\,p+9\,a\,b^2\,d^2\,p\,x\right )+a^3\,e^4\,p^2+9\,a^2\,b\,d^3\,e\,p^2+6\,a^2\,b\,d^2\,e^2\,p^2\,x\right )\,\mathrm {root}\left (27\,b^3\,c^3-27\,a\,b^2\,c^2\,e^2\,p+81\,a\,b^2\,c\,d^3\,e\,p^2+9\,a^2\,b\,c\,e^4\,p^2-27\,a\,b^2\,d^6\,p^3-a^3\,e^6\,p^3,c,k\right )\right )+\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,\left (d^2\,x+d\,e\,x^2+\frac {e^2\,x^3}{3}\right )-3\,d^2\,p\,x-\frac {e^2\,p\,x^3}{3}-\frac {3\,d\,e\,p\,x^2}{2} \]

[In]

int(log(c*(a + b*x^3)^p)*(d + e*x)^2,x)

[Out]

symsum(log(root(27*b^3*c^3 - 27*a*b^2*c^2*e^2*p + 81*a*b^2*c*d^3*e*p^2 + 9*a^2*b*c*e^4*p^2 - 27*a*b^2*d^6*p^3
- a^3*e^6*p^3, c, k)*(9*root(27*b^3*c^3 - 27*a*b^2*c^2*e^2*p + 81*a*b^2*c*d^3*e*p^2 + 9*a^2*b*c*e^4*p^2 - 27*a
*b^2*d^6*p^3 - a^3*e^6*p^3, c, k)*a*b^2 - 6*a^2*b*e^2*p + 9*a*b^2*d^2*p*x) + a^3*e^4*p^2 + 9*a^2*b*d^3*e*p^2 +
6*a^2*b*d^2*e^2*p^2*x)*root(27*b^3*c^3 - 27*a*b^2*c^2*e^2*p + 81*a*b^2*c*d^3*e*p^2 + 9*a^2*b*c*e^4*p^2 - 27*a
*b^2*d^6*p^3 - a^3*e^6*p^3, c, k), k, 1, 3) + log(c*(a + b*x^3)^p)*(d^2*x + (e^2*x^3)/3 + d*e*x^2) - 3*d^2*p*x
- (e^2*p*x^3)/3 - (3*d*e*p*x^2)/2